Quoting and citing are two different things and the commands used for the two jobs are mostly independent.
Citing
I wrote about the 'evolution' of \cite and related commands in A design question: citation commands. Roughly speaking you could say that originally LaTeX only had \cite and that new commands were added to new packages as people realised that other citation commands would be useful and found a way to implement them.
Nowadays you probably have a choice between vanilla LaTeX's \cite, cite, natbib and biblatex (there are other specialised packages such as apacite, jurabib, ...). As a general rule you can only load one of the three packages cite, natbib and biblatex in your document. Each package has its advantages and disadvantages and there are many comparisons on this site that you could consult for your decision. bibtex vs. biber and biblatex vs. natbib
If you are using biblatex I suggest you try and use \autocite as your go-to citation command. \autocite is intended to be flexible and allow for easy changes of your citation style.
Very related: Universal `\cite` commands or defining new cite commands.
Quoting
Standard LaTeX defines the quote and quotation environments for longer (indented) block quotes of text. The quoting defines the environment quoting that is slightly more flexible than the standard LaTeX environments.
Then there is csquotes (by the same author as biblatex), which defines commands for short citations of a few words, longer block-quote citations and even has commands that allow you to combine quoting and citing into one (\textcquote, \blockcquote, ...). Have a look at the documentation for all commands csquotes defines (I will show some important commands below).
Very related: "direct" quotations and "entire paragraph" quotations.
Example
biblatex's \autocite and some csquotes commands have limited ability to move around punctuation marks. In almost all cases this can only happen if the punctuation mark comes after the command in question. But there is no command that knows if your citation is relevant only for a sentence or the entire paragraph. So in the end you have to decide where the punctuation should go.
\documentclass[ngerman]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[style=authoryear, backend=biber]{biblatex}
\usepackage{kantlipsum}
\addbibresource{biblatex-examples.bib}
\SetCiteCommand{\autocite} % tell csquotes to use biblatex's \autocite for citations
\begin{document}
Lorem ipsum dolor sit amet \autocite[380]{sigfridsson}.
Lorem ipsum dolor sit amet. \autocite[380]{sigfridsson}
Lorem ipsum \enquote{dolor} sit amet.
Lorem ipsum \textcquote[380]{sigfridsson}{dolor} sit amet.
\blockquote{\kant[1-2]}
Lorem ipsum dolor
\blockcquote[18]{kant:kpv}{\kant*[1]\par\kant*[2]}
\blockquote{Lorem ipsum dolor}
\blockcquote[18]{kant:kpv}{Lorem ipsum dolor}
\printbibliography
\end{document}
